Question

A mass-spring-dashpot system is described by

my′′ + cy′ + ky = F_o cos ωt,

see §3.6 Eq. (17). This second-order differential equation has been used in simulations, such as this

one at the PhET site: https://phet.colorado.edu/en/simulation/legacy/resonance.

For m = 2.53kg, c = 0.502N/(m/s), k = 97.2N/m, F_o = 97.2×0.5N = 48.6N,and ω = 2.6, the equation becomes

2.53y′′ + 0.502y′ + 97.2y = 48.6 cos(ωt) ....................... (1)

(a) Given the initial value

y(0) = 2.20, y′(0) = 0,

solve Eq. (1), Round to three significant figures. Show all your work, and clearly highlight your

conclusion—the combination of complementary function and particular solution.

(b) From your particular solution, which is of the form

y_p =Acosωt+Bsinωt,

calculate the amplitude, which is √(A2 + B2). Clearly highlight your conclusion—the amplitude. You are encouraged to use the PhET simulation to verify your amplitude. Note that the angular frequency ω = 2πf where f is the frequency in the simulation.

Answer 1